When ds was first applying to a U.S. private school, my request for a recommendation was initially a straight No. Our Canadian public school teachers had it in their contract that they would not write recommendations for private schools. Ultimately, I had the admission officer and principal set up a phone call and that worked for the private school.
Some of the recommendation forms get very detailed. As a teacher, I would not be comfortable answering many of the questions, like how they get along with other students, if I had only seen the student in an online format.